# Intertemporal consumer choice

## Discounted utility

### Discounting

Time inconsistency (hyperbolic discounting). again, this should mirror ai. page overview if necessary. call it that?

## Intertemporal budget constraint

### Budget constraint

$$\sum_{t=T}C^t(1+r)^{-t}=\sum_{t=T}Y_t(1+r_t)^{-t}+W_T$$

$$W_t$$ is wealth endowment at time $$T$$.

### The Euler equation

If we have exponential discounting we have:

$$U_T=E[ \sum_{t=T}^\infty (1+\delta )^t U(C_t) ]$$

The first-order conditions give us:

$$u'(x_t)=(1+\delta)(1+r_t)u'(x_{t+1})$$

### The Euler equation with liquidity constraints

$$u'(x_t)=(1+\delta)(1+r_t)u'(x_{t+1})+\lambda_{t+1}$$

## Household finance

### Household wealth and liquidity

Hold cash, equity, bonds, mortgages.

## Other

### Habit formation

Different utility function.

### Durable goods

Can wait to purchase. depends on expected prices in future.

### Buying or owning

production of other commodities. house produces rentable space eg if you own house, you have rentable space each period.

depreciation? 1 for something like rent, maybe 0.01 for long term asset